Religion stands on the foundation of acceptance by faith or without tangible/solid evidence. Science takes on the precise opposite-- acceptance of ideas/laws/concepts as based upon fact, experimentation, and observation. Furthermore, science takes on a progressive stance in which incorrect or flawed theories may be DEVELOPED or even completely disregarded as new evidence is brought up; science doesn't stand still. Mainstream religion portrays itself as perfect (despite glaringly obvious signs of the contrary) and thus refuses most fundamental change (due in part to the assertion that it cannot be improved, it is already "complete"). The vast majority of religious individuals (virtually all) raise or will raise their children to adhere to the rules and values of their respective religion before they are even able to read and write which instills a deeply ingrained bias towards their religion and often a harrowing disregard (which often escalates to a hatred) for other religions and foreign ideas they come across during their lifetime. A scientist (and science as a whole) accepts new ideas provided they are better or add significantly to previous theories.

If you mean incompatible

Then yes, religion and science are fundamentally incompatible. Science explains the natural world not the supernatural world. Because science is best defined by scientific methodology, an accurate understanding of science also means understanding why science is superior to faith, intuition, or any other method of acquiring knowledge. Religion, true to its contradicting ways, both embraces and denounces science. Science never uses religion to advance its position. Religion may need science, but science will never need religion.By every reasonable measure science has improved the lot of humanity by orders of magnitude more than any other institution. The scientific method is the most important discovery in the entire history of the human race. The essence of that method is actively seeking out disconfirming evidence, doubt, analysis, critique, skepticism, and defeasibility. The essence of religion dogmatism, belief, authority, stagnation, rationalizing, and conformity. Religion is a social institution that propagates the view that disagreeing with it is disrespectful, intolerant, sinful, and evil.

Their not enemies, but forces that counter act each other

Science and religion are not enemies, but they disagree with each other a lot of times. For example lets use galileo in your argument. He had supported the heliocentric theory, which says the sun in the middle, however this countered the church's geocentric theory, earth based theory. Then the church had threatened galileo with heresy. Then Galileo recanted his statements. This is just one of many examples. A movie that will address your debate is God's not Dead.

Only in the sense that they can't occupy the same space.

Religion was the science before science existed. It answered all the questions we had about our world to the best of it's ability. Religion will be phased out, granted it will take many more years for this to happen. The more science unveils the less religion can declare without evidence.

The ideologies presented by mainstream religion are contradictory to those presented by science and the scientific method

Religion stands on the foundation of acceptance by faith or without tangible/solid evidence. Science takes on the precise opposite-- acceptance of ideas/laws/concepts as based upon fact, experimentation, and observation. Furthermore, science takes on a progressive stance in which incorrect or flawed theories may be DEVELOPED or even completely disregarded as new evidence is brought up; science doesn't stand still. Mainstream religion portrays itself as perfect (despite glaringly obvious signs of the contrary) and thus refuses most fundamental change (due in part to the assertion that it cannot be improved, it is already "complete"). The vast majority of religious individuals (virtually all) raise or will raise their children to adhere to the rules and values of their respective religion before they are even able to read and write which instills a deeply ingrained bias towards their religion and often a harrowing disregard (which often escalates to a hatred) for other religions and foreign ideas they come across during their lifetime. A scientist (and science as a whole) accepts new ideas provided they are better or add significantly to previous theories.

Science and religion are not enemies, just different languages telling the same story. Science was branded a foe by religion hundreds of years ago when it felt threatened, and in a misguided attempt to maintain control over the public, the Church persecuted scientists like Newton and Galileo and suppressed any teaching that does not the same as theirs.

They don't work together.

Religion and science are fundamentally incompatible. Science explains the natural world not the supernatural world. Because science is best defined by scientific methodology, an accurate understanding of science also means understanding why science is superior to faith, intuition, or any other method of acquiring knowledge. Religion, true to its contradicting ways, both embraces and denounces science. Science never uses religion to advance its position. Religion may need science, but science will never need religion.By every reasonable measure science has improved the lot of humanity by orders of magnitude more than any other institution. The scientific method is the most important discovery in the entire history of the human race. The essence of that method is actively seeking out disconfirming evidence, doubt, analysis, critique, skepticism, and defeasibility. The essence of religion dogmatism, belief, authority, stagnation, rationalizing, and conformity. Religion is a social institution that propagates the view that disagreeing with it is disrespectful, intolerant, sinful, and evil.

No they aren't.

Science and religion are not mutually exclusive. Science looks more at the 'how' type of questions, and in the answers to the 'how' questions are tools which make certain phenomena in the universe predictable. If they do not make certain aspects of the universe predictable they no longer answer the 'how' question and it is left open to anyone anywhere to come up with a satisfactory answer that allows people to make predictions. Science is filled with 'how' questions, and corresponding answers which allow for predictions. For example:

How do the planets move? A: Keplar's Laws of Planetary Motion, which allow you to predict where the planets will be in the future.

How do voltage, resistance, and current interact electrically? A: Ohm's Law, which allows you to design electronic circuits.

How do the particles in the universe behave at a subatomic scale? A: We don't know yet, but our current understanding led us to predict the existence of the Higgs Boson in 1964, which was finally confirmed in 2012.

I can go on and on.

Religion however, in my opinion, is more concerned with answering from a more philosophical perspective, the big 'Why?' question. Which results in questions like: "Why am I here?" "Who is my creator? (assuming there is one)" etc. Like scientific theories, not all of them can be correct since many are competing and conflicting at the same time. This is where the conflict arises. And many can be conflicting with science, but that does not make them enemies.

They don't have to be.

Religion is primarily a set of moral conducts often borne from history and followed by a number of people. Science is something that can be isolated from religion. There can exist a happy medium where people study life and beyond without having a religious crisis, but by adapting to the things they know and forming conceptions about how their religion fits in with it.

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They go hand-in-hand.

Science and Christianity support each other in a vast amount of ways. Science gave a name to "Let there be Light." they called it the Big Bang. Astronomy affirms the account in Job where it says the universe is being stretched out, and that Earth is suspended over nothing. Job also says the Earth is round, both confirmed by science. The Bible talks about the pattern of the wind, it describes dinosaurs (this is also in Job.) Dr. C. Venter concluded that all human genomes are so similar, we must be of the same genealogy. The Bible is full of facts that science has proved in recent years, and will continue to be proven.